Problem: All of the 3rd grade teachers and students from Almond went on a field trip to an archaeology museum. Tickets were $$7.50$ each for teachers and $$4.50$ each for students, and the group paid $$66.00$ in total. The next month, the same group visited a science museum where the tickets cost $$22.50$ each for teachers and $$11.00$ each for students, and the group paid $$178.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+4.5y = 66}$ ${22.5x+11y = 178}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-22.5x-13.5y = -198}$ ${22.5x+11y = 178}$ Add the top and bottom equations together. $ -2.5y = -20 $ $ y = \dfrac{-20}{-2.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {7.5x+4.5y = 66}$ to find $x$ ${7.5x + 4.5}{(8)}{= 66}$ $7.5x+36 = 66$ $7.5x = 30$ $x = \dfrac{30}{7.5}$ ${x = 4}$ You can also plug ${y = 8}$ into $ {22.5x+11y = 178}$ and get the same answer for $x$ ${22.5x + 11}{(8)}{= 178}$ ${x = 4}$ There were $4$ teachers and $8$ students on the field trips.